General Trees. Tree Traversal Algorithms.

• GeneralTrees.• TreeTraversalAlgorithms.• BinaryTrees.

2CPSC 3200 University of Tennessee at Chattanooga – Summer 2013 © 2010 Goodrich, Tamassia

• Incomputerscience,atreeisanabstractmodelofahierarchicalstructure.

• Atreeconsistsofnodeswithaparent-childrelation.

• Applications:• Organizationcharts.• Filesystems.• Programmingenvironments.

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CPSC 3200 University of Tennessee at Chattanooga – Summer 2013 3© 2010 Goodrich, Tamassia

Subtree

• Root:nodewithoutparent(A)• Internalnode: nodewithatleastonechild(A,B,C,F)

• Externalnode(a.k.a.leaf):nodewithoutchildren(E,I,J,K,G,H,D)

• Ancestorsofanode:parent,grandparent,grand-grandparent,etc.

• Depthofanode:numberofancestors

• Heightofatree:maximumdepthofanynode(3)

• Descendantofanode:child,grandchild,grand-grandchild,etc.

A

B DC

G HE F

I J K

• Subtree:treeconsistingofanodeanditsdescendants.


• edgeoftreeTisapairofnodes(u,v)suchthatuistheparentofv,orviceversa.

• PathofTisasequenceofnodessuchthatanytwoconsecutivenodesinthesequenceformanedge.

• Atreeisordered ifthereisalinearorderingdefinedforthechildrenofeachnode


• Weusepositions(nodes)toabstractnodes.• getElement(): Returntheobjectstoredatthisposition.

• Genericmethods:• integergetSize()• boolean isEmpty()• Iterator iterator()• Iterable positions()

• Accessor methods:• positiongetRoot()• positiongetThisParent(p)• Iterable children(p)

• Querymethods:• boolean isInternal(p)• boolean isExternal(p)• boolean isRoot(p)

• Updatemethod:• elementreplace(p,o)

• AdditionalupdatemethodsmaybedefinedbydatastructuresimplementingtheTreeADT.


• LetvbeanodeofatreeT.Thedepth ofv isthenumberofancestorsofv,excludingv itself.• Ifv istheroot,thenthedepthofv is0• Otherwise,thedepthofv isoneplusthedepthoftheparentofv.

• Therunningtimeofalgorithmdepth(T,v)isO(dv),wheredvdenotesthedepthofthenodev inthetreeT.

CPSC 3200 University of Tennessee at Chattanooga – Summer 2013 8

Algorithm depth(T,v):if v istherootofTthen

return 0else

return 1+depth(T,w),wherewistheparentofv inT

© 2010 Goodrich, Tamassia

• Atreeisadatastructurewhichstoreselementsinparent-child relationship.

A

B C

D E F G H

Root node

Internal nodes

Leaf nodes (External nodes)

Siblings

Siblings

Siblings


• Depth:thenumberofancestorsofthatnode(excludingitself).

• Height: themaximumdepthofanexternalnodeofthetree/subtree.

A

B C

D E F G H

I

Depth(D) = ?Depth(D) = 1Depth(D) = 2

Depth(I) = ?Depth(I) = 3

Height = MAX[ Depth(A), Depth(B), Depth(C), Depth(D), Depth(E), Depth(F), Depth(G), Depth(H), Depth(I) ]

Height = MAX[ 0, 1, 1, 2, 2, 2, 2, 2, 3 ] = 3CPSC 3200 University of Tennessee at Chattanooga – Summer 2013 10

• Theheight ofanodev inatreeTiscanbecalculatedusingthedepth algorithm.

• algorithmheight1 runsinO(n2)time


Algorithmheight1(T):h←0for eachvertexv inTdo

if v isanexternalnodeinTthenh←max(h,depth(T,v))

return h


• Theheight ofanodev inatreeTisalsodefinedrecursively:• Ifv isanexternalnode,thentheheightofv is0• Otherwise,theheightofv isoneplusthemaximumheightofachildofv.

• algorithmheight1 runsinO(n)timeCPSC 3200 University of Tennessee at Chattanooga – Summer 2013 12

Algorithmheight2(T,v):ifv isanexternalnodeinTthen

return 0else

h←0for eachchildwofv inTdo

h←max(h,height2(T,w))return 1+h


• Atraversalvisitsthenodesofatreeinasystematicmanner.

• Inapreorder traversal,anodeisvisitedbeforeitsdescendants.

• Application:printastructureddocument.

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Algorithm preOrder(v)visit(v)for each childw ofv

preorder (w)


• Inapostorder traversal,anodeisvisitedafteritsdescendants.

• Application:computespaceusedbyfilesinadirectoryanditssubdirectories.

Algorithm postOrder(v)for each childw ofv

postOrder (w)visit(v)

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• Theorderinwhichthenodesarevisitedduringatreetraversalcanbeeasilydeterminedbyimaginingthereisa“flag”attachedtoeachnode,asfollows:

• Totraversethetree,collecttheflags:

preorder inorder postorder

A

B C

D E F G

A

B C

D E F G

A

B C

D E F G

A B D E C F G D B E A F C G D E B F G C ACPSC 3200 University of Tennessee at Chattanooga – Summer 2013 15

• Theothertraversalsarethereverseofthesethreestandardones• Thatis,therightsubtree istraversedbeforetheleftsubtree istraversed

• Reversepreorder: root,rightsubtree,leftsubtree.• Reverseinorder:rightsubtree,root,leftsubtree.• Reversepostorder: rightsubtree,leftsubtree,root.


• Abinarytreeisatreewiththefollowingproperties:• Eachinternalnodehasatmosttwochildren (exactlytwoforproperbinarytrees).• Thechildrenofanodeareanorderedpair.

• Wecallthechildrenofaninternalnodeleftchild andrightchild.

• Alternativerecursivedefinition:abinarytreeiseither• atreeconsistingofasinglenode,or• atreewhoseroothasanorderedpairofchildren,eachofwhichisabinarytree.

A

B C

F GD E

H I

Applications:• arithmeticexpressions.• decisionprocesses.• searching.


• Abinarytreeisbalancedifeverylevelabovethelowestis“full”(contains2h nodes)

• Inmostapplications,areasonablybalancedbinarytreeisdesirable.

a

b c

d e f g

h i jA balanced binary tree

a

b

c

d

e

f

g h

i jAn unbalanced binary tree


• Binarytreeassociatedwithadecisionprocess• internalnodes:questionswithyes/noanswer• externalnodes:decisions

• Example:diningdecision

Want a fast meal?

How about coffee? On expense account?

Starbucks Spike’s Al Forno Café Paragon

Yes No

Yes No Yes No


• Binarytreeassociatedwithanarithmeticexpression• internalnodes:operators• externalnodes:operands

• Example: arithmeticexpressiontreefortheexpression(2´ (a- 1)+ (3´ b))

+

´´

-2

a 1

3 b


• Isabinarytreewherethenumberofexternalnodesis1morethanthenumberofinternalnodes.



A

B C

D

Internal nodes = 2External nodes = 2



A

B C

D


E



A

B C

D


E F



A

B C

D


E F G


Worstcase: Thetreehavingtheminimumnumberofexternalandinternalnodes.

Bestcase: Thetreehavingthemaximumnumberofexternalandinternalnodes.

1.Thenumberofexternalnodesisatleasth+1 andatmost2h

Ex:h=3

External nodes = 3+1 = 4

External nodes = 23 = 8


2.Thenumberofinternalnodesisatleasth andatmost2h-1Ex:h=3Worstcase: Thetreehavingtheminimumnumberofexternalandinternalnodes.

Bestcase: Thetreehavingthemaximumnumberofexternalandinternalnodes.

Internal nodes = 3

Internal nodes = 23 -1=7


3.Thenumberofnodesisatleast2h+1 andatmost2h+1 -1Ex:h=3


----------------------------Internal + External = 2*3 +1 = 7

Internal nodes = 7External nodes = 8-----------------------

Internal + External = 23+1 – 1 = 15


4.Theheightisatleastlog(n+1)-1andatmost(n-1)/2

Number of nodes = 7h = 3

Number of nodes = 15h = 3


• TheBinaryTree ADTextends theTreeADT,i.e.,itinheritsallthemethodsoftheTreeADT.

• Additionalmethods:• positiongetThisLeft(p)• positiongetThisRightight(p)• boolean hasLeft(p)• boolean hasRight(p)

• UpdatemethodsmaybedefinedbydatastructuresimplementingtheBinaryTree ADT.


• Anodeisrepresentedbyanobjectstoring• Element• Parentnode• Leftchildnode• Rightchildnode

• NodeobjectsimplementthePositionADT

B

DA

C E

Æ Æ

Æ Æ Æ Æ

B

A D

C E

Æ


• addRoot(e): Createandreturnanewnoderstoringelementeandmakertherootofthetree;anerroroccursifthetreeisnotempty.

• insertLeft(v,e): Createandreturnanewnodewstoringelemente,addwasthethe leftchildofvandreturnw;anerroroccursifvalreadyhasaleftchild.

• insertRight(v,e): Createandreturnanewnodezstoringelemente,addzasthethe rightchildofvandreturnz;anerroroccursifvalreadyhasarightchild.

• remove(v): Removenodev,replaceitwithitschild,ifany,andreturntheelementstoredatv;anerroroccursifvhastwochildren.

• attach(v,T1,T2): AttachT1andT2,respectively,astheleftandrightsubtrees oftheexternalnodev;anerrorconditionoccursifv isnotexternal.


• Binarytreesareexcellentdatastructuresforsearchinglargeamountsofinformation.

• Whenusedtofacilitatesearches,abinarytreeiscalledabinarysearchtree.


• Abinarysearchtree(BST)isabinarytreeinwhich:• Elementsinleft subtree aresmaller thanthecurrentnode.• Elementsinright subtree aregreater thanthecurrentnode.

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7 12

5 9 11 25


• Therearethreecommonmethodsfortraversingabinarytreeandprocessingthevalueofeachnode:• Pre-order• In-order• Post-order

• Eachofthesemethodsisbestimplementedasarecursivefunction.


• Pre-order: Nodea Lefta Right

A

B C

D E F G

A B D E C F GCPSC 3200 University of Tennessee at Chattanooga – Summer 2013 37

• Insertthefollowingitemsintoabinarysearchtree.50,25,75,12,30,67,88,6,13,65,68

• DrawthebinarytreeandprinttheitemsusingPre-ordertraversal.


• In-order: Lefta Nodea Right

A

B C

D E F G

D B E A F C GCPSC 3200 University of Tennessee at Chattanooga – Summer 2013 39

• Fromthepreviousexercise,printthetree’snodesusingIn-ordertraversal.

50

25 75

12 30 67 88

6 13 65 68


• Post-order: Lefta Righta Node

A

B C

D E F G

D E B F G C ACPSC 3200 University of Tennessee at Chattanooga – Summer 2013 41

• Fromthepreviousexercise,printthetree’snodesusingPost-ordertraversal.

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25 75

12 30 67 88

6 13 65 68


• Inaninorder traversalanodeisvisitedafteritsleftsubtreeandbeforeitsrightsubtree

• Application:drawabinarytree• x(v)=inorder rankofv• y(v)=depthofv

Algorithm inOrder(v)if hasLeft (v)

inOrder (left (v))visit(v)if hasRight (v)

inOrder (right (v))

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• Afterdeletinganitem,theresultingbinarytreemustbeabinarysearchtree.1. Findthenodetobedeleted.2. Deletethenodefromthetree.


• Thenodetobedeletedhasnoleftandrightsubtree (thenodetobedeletedisaleaf).

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50 70

30 53 65 80

51 57 61 67 79 95

delete(30)


• Thenodetobedeletedhasnoleftsubtree (theleftsubtreeisemptybutithasanonemptyrightsubtree).

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50 70

30 53 65 80

35 51 57 61 67 79 95

delete(30)


• Thenodetobedeletedhasnorightsubtree (therightsubtree isemptybutithasanonemptyleftsubtree).

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50 70

30 53 65 80

25 35 51 57 61 67 79

delete(80)


• Thenodetobedeletedhasnonemptyleftandrightsubtree.

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50 70

30 53 65 80

25 35 51 57 61 67 79 95

delete(70)

79


• Thenodetobedeletedhasnonemptyleftandrightsubtree.

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50 70

30 53 65 80

25 35 51 57 61 67 79 95

delete(70)

67


50

• Binarysearchcanperformoperationsget,floorEntry andceilingEntry onanorderedmapimplementedbymeansofanarray-basedsequence,sortedbykey• similartothehigh-lowgame• ateachstep,thenumberofcandidateitemsishalved• terminatesafterO(logn)steps

• Example:find(7)1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

0

0

0

0

ml h

ml h

ml h

l=m =hCPSC 3200 University of Tennessee at Chattanooga – Summer 2013 © 2010 Goodrich, Tamassia

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• Abinarysearchtreeisabinarytreestoringkeys(orkey-valueentries)atitsinternalnodesandsatisfyingthefollowingproperty:• Letu,v,andw bethreenodessuchthatu isintheleftsubtree ofv andw isintherightsubtreeofv.Wehavekey(u)£ key(v)£ key(w)

• Externalnodesdonotstoreitems.

• Aninorder traversalofabinarysearchtreesvisitsthekeysinincreasingorder.

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CPSC 3200 University of Tennessee at Chattanooga – Summer 2013 © 2010 Goodrich, Tamassia

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• Tosearchforakeyk,wetraceadownwardpathstartingattheroot.

• Thenextnodevisiteddependsonthecomparisonofkwiththekeyofthecurrentnode.

• Ifwereachaleaf,thekeyisnotfound.

• Example:get(4):• CallTreeSearch(4,root)

Algorithm TreeSearch(k, v)if T.isExternal (v)return v

ifk < key(v)return TreeSearch(k, T.left(v))

elseifk = key(v)return v

else {k > key(v)}return TreeSearch(k, T.right(v))

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CPSC 3200 University of Tennessee at Chattanooga – Summer 2013 © 2010 Goodrich, Tamassia

General Trees. Tree Traversal Algorithms.

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